This paper reports a search for Higgs boson pair (hh) production in association with a vector boson (W or Z) using 139 $fb^{-1}$ of the proton-proton collision data at $\sqrt{s}$ = 13 TeV recorded with the ATLAS detector at the Large Hadron Collider. The search is performed in final states in which the vector boson decays leptonically (W$\rightarrow lv$, Z$\rightarrow ll,vv$ with $l$ = e, µ) and the Higgs bosons decay to a pair of b-quarks. It targets Vhh signals from both the Standard Model (SM) like non-resonant hh production and the resonant hh production predicted in some SM extensions. A 95% confidence-level upper limit of 183 (87) times the SM cross-section is observed (expected) for non-resonant V hh production when assuming SM-like kinematics. Constraints on Higgs boson coupling modifiers are also set. For the resonant search, upper limits on the production cross-sections are derived for two specific models: one is the production of a vector boson along with a neutral heavy scalar resonance H in the mass range 260 - 1000 GeV that decays into hh, and the other is the production of a heavier neutral pseudoscalar resonance A that decays into a Z and H boson in the mass range 360 - 800 GeV for the A boson and 260 - 400 GeV for the H boson. Constraints in the parameter space of two-Higgs-doublet models are also derived.

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